## Factoring over

In other words, we shall express as a product of irreducible polynomials having coeficients in .

This is, in fact, very easy. The complex roots of are precisely the roots of unity: those numbers for which

To determine these, recall that any complex number may be written in its polar decomposition, , where and are it's polar coordinates''. (Moreover, if then these coordinates are unique.) Then implies , which in turn implies (so since ) and . Substituting these into , we find that must be equal to one of the following numbers:

In fact, this sequence is periodic, so only the first terms are distinct and after that the numbers start repeating themselves. We summarize this as the following result.

Lemma 2.10.5   The complex roots of are

These may be visualized as follows. In general, plot the complex number

at the point in the -plane. (Sometimes this way of plotting complex numbers is called the Gaussian plane.) The roots of are equally spaced points on the unit circle, starting at .

In fact,

is the factorization into irreducibles opver . In other words, factoring over amounts geometrically to subdividing the circle into equal parts.

Remark 2.10.6   The ancient Greeks asked for which is it possible to subdivide the circle into equal parts using only a ruler and compass. This, and its connection with factoring polynomials, is discussed in further detail in chapter 22 of Schroeder [Sch].

david joyner 2008-04-20